Linearized dynamical system for extended magnetohydrodynamics in terms of Lagrangian displacement fields and isovortical perturbations

Linear stability theory for extended magnetohydrodynamics (XMHD), which incorporates the Hall and electron-inertia effects into MHD, is developed by introducing Lagrangian displacement fields for ions and electrons. For simplicity, incompressible and dissipationless XMHD is assumed in this work, since the present consideration is greatly inspired by an analogy with incompressible ideal fluid. The governing equations for the Lagrangian displacement fields are written as a gyroscopic dynamical system, to which the well-established canonical Hamiltonian theory is ready to apply. This linear perturbation can be further restricted to the isovortical one in the same manner as ideal fluid, which is more generally understood as dynamically accessible perturbation in a constrained Hamiltonian system. In XMHD, there are two isovortical constraints corresponding to the two canonical vorticities for ions and electrons. It is shown that the Frieman-Rotenberg equation for ideal MHD is reproduced by not only neglecting the Hall and electron-inertia effects but also imposing one of the two isovortical constraints. As an application, a new stability condition for static equilibria of inertial MHD is derived.